ENTROPY FORMULATION FOR PARABOLIC DEGENERATE EQUATIONS...

ENTROPY FORMULATION FOR PARABOLIC DEGENERATEEQUATIONS WITH GENERAL DIRICHLET BOUNDARY

CONDITIONS AND APPLICATION TO THE CONVERGENCE OFFV METHODS∗

ANTHONY MICHEL† AND JULIEN VOVELLE‡

SIAM J. NUMER. ANAL. c© 2003 Society for Industrial and Applied MathematicsVol. 41, No. 6, pp. 2262–2293

Abstract. This paper is devoted to the analysis and the approximation of parabolic hyperbolicdegenerate problems defined on bounded domains with nonhomogeneous boundary conditions. Itconsists of two parts. The first part is devoted to the definition of an original notion of entropysolutions to the continuous problem, which can be adapted to define a notion of measure-valuedsolutions, or entropy process solutions. The uniqueness of such solutions is established. In thesecond part, the convergence of the finite volume method is proved. This result relies on (weak)estimates and on the theorem of uniqueness of the first part. It also entails the existence of asolution to the continuous problem.

Key words. parabolic degenerate equations, boundary conditions, finite volume methods

AMS subject classifications. 35K65, 35F30, 35K35, 65M12

DOI. 10.1137/S0036142902406612

1. Introduction. Let Ω be an open bounded polyhedral subset of Rd and T ∈

R∗+. Let us denote by Q the set (0, T )× Ω, and by Σ the set (0, T )× ∂Ω.We consider the following parabolic-hyperbolic problem:

ut + div(F (t, x, u))−∆ϕ(u) = 0, (t, x) ∈ Q,u(0, x) = u0(x), x ∈ Ω,u(t, x) = u(t, x), (t, x) ∈ Σ .

(1)

Such an equation of quasilinear advection with degenerate diffusion governs the evolu-tion of the saturation of the wetting fluid in the study of diphasic flow in porous media[GMT96], [Mic01], [EHM01]. In that case, the function ϕ can be expressed using thecapillary pressure and the relative mobilities. The function ϕ is only supposed to bea nondecreasing Lipschitz continuous function. In particular, the study of problem(1) includes the study of nonlinear hyperbolic problems (cases where ϕ′ = 0).

The analysis of the approximation of nonlinear hyperbolic problems via the finitevolume (FV) method began in the mid 1980s, involving several authors including,for example, Cockburn, Coquel, and LeFloch [CCL95], Szepessy [Sze91], Vila [Vil94],Kroner, Rokyta, and Wierse [KRW96], and Eymard, Gallouet, and Herbin [EGH00].Results on the convergence of FV schemes for degenerate problems in general cameto light in more recent years [EGHM02], [Ohl01]. See also [BGN00], [EK00] for othermethods of approximation.

When the function ϕ is strictly increasing, problem (1) is of parabolic type. Inthat case, the existence of a unique weak solution is well known. In the case whereϕ′ = 0, problem (1) is a nonlinear hyperbolic problem, the uniqueness of a weak

∗Received by the editors April 29, 2002; accepted for publication (in revised form) April 1, 2003;published electronically December 5, 2003.

http://www.siam.org/journals/sinum/41-6/40661.html†Departement de Mathematiques, Universite de Montpellier II, CC 051–Place Eugene Bataillon,

F-34095 Montpellier cedex 5, France ([email protected]).‡Universite de Provence, Centre de Mathematiques et d’Informatique, F-13453 Marseille, France

([email protected]).

2262

DEGENERATE PARABOLIC EQUATION ON BOUNDED DOMAIN 2263

solution is not ensured, and one has to define a notion of entropy solutions to recoveruniqueness [Kru70]. Therefore, it is quite difficult to define a notion of solution in thecase where ϕ is merely a nonincreasing function. In fact, as far as the Cauchy problemin the whole space is concerned, such a definition has been done for a long time, sinceVolpert and Hudjaev [VH69], but uniqueness with nonlinear parabolic terms has onlybeen proved recently by Carrillo [Car99] (see also [KO01], [KR00]).

Another difficulty in the study of degenerate parabolic problems is analysis of theboundary conditions (see [LBS93], [RG99]). It is not always easy to give a correctformulation of the boundary conditions, or of the way they have to be taken intoaccount. In the case where the function ϕ is strictly increasing, the classical frameworkof variational solutions of parabolic equations is enough to satisfy this wish. In thecase where ϕ′ = 0, things are completely different. Even if the (entropy) solution uof problem (1) admits a trace (say, γu) on Σ, the equality γu = u on Σ does notnecessarily hold. Actually, a condition on Σ can be given, which is known as theBLN condition [BLN79]: this is the right way to formulate boundary conditions inthe study of scalar hyperbolic problems. However, the notion of entropy solutionto nonlinear Cauchy–Dirichlet hyperbolic problems given by Bardos, LeRoux, andNedelec is not really suitable to the study of FV schemes since it requires that thesolution u be in a space BV (because the trace of u is involved in the formulation ofthe BLN condition), and it is known that it is difficult to get BV estimates on thenumerical approximations given by the FV method on non-Cartesian grids. Actually,Otto gave an integral formulation of entropy solutions to scalar hyperbolic problemswith boundary conditions [Ott96], and this indeed allows us to prove the convergenceof the FV method [Vov02].

To our knowledge, the problem that we deal with (convergence of the FV methodfor degenerate parabolic equations with nonhomogeneous boundary conditions) hasnever been considered before. Nevertheless, in [MPT02], the authors give a definitionof entropy solution for which uniqueness and consistency with the parabolic approxi-mation are proved. This definition is not completely in integral form and therefore notsuitable for proving the convergence of the FV method, since only poor compactnessresults are available on the numerical approximation. That is why we give an origi-nal definition of the problem (see Definition 3.1). This complete integral formulationincludes the definition of Otto but not exactly the one of Carrillo (see the commentsthat follow Definition 3.1). It is well suited to the study of the convergence of severalapproximations of problem (1) and is used, for example, in [GMT02] to prove theconvergence of a discrete Bhatnagar–Gross–Krook (BGK) model (see also [MPT02]for the parabolic approximation).

Notice that some particular cases have been fully treated: in [EGHM02], theauthors prove the convergence of the FV method in the case where F (x, t, s) =q(x, t)f(s), div(q) = 0, with q · n = 0 on Σ. In that case, the boundary condi-tion does not act on the hyperbolic part of the equation. From a technical point ofview, this means that the influence of the boundary condition appears in the termsrelated to the parabolic degenerate part of the equation. These parabolic degenerateterms are estimated by following the methods of Carrillo in [Car99], who deals withhomogeneous boundary conditions. On the other hand, in [Vov02], the author provesthe convergence of an FV method in the case where ϕ′ = 0, adapting the ideas of Otto[Ott96]. In that case, the effects of the boundary condition in the hyperbolic equationare the center of the work. In this paper we mix these two precedent approaches todeal with the parabolic degenerate problem with general boundary conditions.

2264 ANTHONY MICHEL AND JULIEN VOVELLE

We will make the following assumptions on the data:

(H1) F : (t, x, s) → F (t, x, s) ∈ C1(R+ × Rd × R) , divxF = 0,

∂F

∂sis locally Lipschitz continuous uniformly with respect to (t, x);

(H2) ϕ : s → ϕ(s) is a nondecreasing Lipschitz continuous function;

(H3) u0 : x → u0(x) ∈ L∞(Ω); and(H4) the function u : (x, t) → u(x, t) ∈ L∞(Σ) and is the trace of a function

u ∈ L∞(Q)with ϕ(u) ∈ L2(0, T ;H1(Ω)) .

To prove the convergence of the FV method, we will also assume that the bound-ary datum satisfies

(H5) the function u : (x, t) → u(x, t) ∈ L∞(Σ) and is the trace of a functionu ∈ L∞(Q) with ϕ(u) ∈ L2(0, T ;H1(Ω)) , ∇u ∈ L2(Q) , ut ∈ L1(Q) .

In the course of the proof of uniqueness of the entropy process solution (Theorem4.1), additional hypotheses on the boundary datum are required. Using the notationdefined in subsection 4.1, they read

(H6) uΣ ∈W 1,1((0, T )×B ∩Q) and ∆ϕ(uΣ) ∈ L1((0, T )×B ∩Q) .Remark 1.1. As suggested by Porretta [MPT02], hypothesis (H6) may be relaxed

as

(H6Bis) uΣ ∈W 1,1((0, T )×B ∩Q) and

∆ϕ(uΣ) is a bounded Radon measure on (0, T )×Π .We do not give a justification of this assertion now. Indeed, hypothesis (H6) is involvedin the proof of Lemma 4.2, and we have waited until Remark 4.1, just after this proof,to specify to what extent hypothesis (H6Bis) is admissible.

Under assumptions (H3)–(H4), there exists (A,B) ∈ R2 such that

A ≤ min(ess inf

Ω(u0), ess inf

Q(u)

)≤ max

(ess sup

Ω(u0), ess sup

Q(u)

)≤ B,(2)

and we set

M = max

∣∣∣∣∂F∂s (t, x, s)∣∣∣∣ , (t, x, s) ∈ Q× [A,B] .

We introduce the function ζ defined by ζ ′ =√ϕ′. (This makes sense in view of (H2).)

We will derive L2(0, T ;H1) estimates on nonlinear quantities such as ζ(u). A simpleexplanation for this fact is the following. Consider the equation ut −∆ϕ(u) = 0 on(0, T ) × Ω. Multiply it by u, and sum the result with respect to x ∈ Ω. The formalidentity

∫Ω∇ϕ(u) · ∇u = ∫

Ω|∇ζ(u)|2 then leads to 1

2ddt

∫Ωu2dx +

∫Ω|∇ζ(u)|2 ≤ 0,

from which can be derived an energy estimate.Notice that the hypothesis divxF = 0 can be relaxed, and source terms can be

considered in the right-hand side of (1).The assumption that u is the trace of an L∞ function u such that ϕ(u) ∈

L2(0, T ;H1(Ω)) is a necessary condition for the existence of solutions to problem (1);


the additional hypotheses introduced in (H5) are involved in the proofs of differentestimates on the approximate solution, defined thanks to the FV method.

As implied at the beginning of this introduction, one of the main points in thestudy of problem (1) is the definition of a notion of solution suitable for the classicaltechniques of convergence of FV schemes. This point is specified in section 2. Insection 3, we introduce and define a notion of entropy process solutions (a conceptsimilar to the concept of measure-valued solutions), and in section 4 we prove theuniqueness of such solutions (see Theorem 4.1). Section 5 is devoted to the FV schemeused to approximate problem (1); a priori estimates are derived and the convergenceis proved.

2. Entropy weak solution. Here, as in the study of purely hyperbolic prob-lems, the concept of weak solutions is not sufficient since the uniqueness of such solu-tions may fail. Thus, we turn to the notion of weak entropy solutions. The entropy-flux pairs considered in the definition of this solution are the so-called Kruzhkov semientropy-flux pairs (η±κ ,Φ

±κ ) (see [Car99], [Ser96], [Vov02]). They are defined by the

formulaη+κ (s) = (s− κ)+ = sκ− κ,η−κ (s) = (s− κ)− = κ− s⊥κ,

Φ+κ (t, x, s) = (s− κ)+ = F (t, x, sκ)− F (t, x, κ),Φ−κ (t, x, s) = (s− κ)− = F (t, x, κ)− F (t, x, s⊥κ),

with ab = max(a, b) and a⊥b = min(a, b). Notice that, in the case where κ is con-sidered as a variable, for example when the doubling variable technique of Kruzhkovis used, the entropy-fluxes will be written

Φ+(x, t, s, κ) = Φ+κ (t, x, s) and Φ−(x, t, s, κ) = Φ−

κ (t, x, s).

Definition 2.1 (entropy weak solution). A function u of L∞(Q) is said to bean entropy weak solution to problem (1) if it is a weak solution of problem (1), thatis, if ϕ(u)− ϕ(u) ∈ L2(0, T ;H1

0 (Ω)) and

∀θ ∈ C∞c ([ 0, T )× Ω) ,(3) ∫Q

u θt + (F (t, x, u)−∇ϕ(u)) · ∇θ dx dt +

∫Ω

u0 θ(0, x) dx = 0,

and if it satisfies the following entropy inequalities for all κ ∈ [A,B], for all ψ ∈C∞c ([ 0, T )× R

d) such that ψ ≥ 0 and sgn±(ϕ(u)− ϕ(κ))ψ = 0 a.e. on Σ:∫Q

η±κ (u)ψt + (Φ±κ (t, x, u)−∇ (ϕ(u)− ϕ(κ))

±) · ∇ψ dx dt +

∫Ω

η±κ (u0)ϕ(0, x) dx

+M

∫Σ

η±κ (u)ψ dγ(x) dt ≥ 0.(4)

Notice that the weak equation (3) is superfluous, for it is a consequence of (4).However, if the function ϕ were (strictly) increasing, (3) would be enough to definea notion of the solution of problem (1) for which existence and uniqueness hold: inthat case, problem (1) would merely be a nonlinear parabolic problem. For generalϕ, the uniqueness of the solution will be a consequence of the entropy inequalities (4);indeed, the class of Kruzhkov semi entropy-flux pairs is wide enough to ensure theuniqueness of the weak entropy solution, while—and we stress this fact—the class ofclassical Kruzhkov entropy-flux pairs s → |s− κ| is not.


Also notice that, first, in the homogeneous case u = 0, the previous definitionis slightly different from the original definition given by Carrillo [Car99] and that,second, if ϕ′ = 0 (problem (1) becomes hyperbolic), then the previous definition ofthe entropy solution coincides with the definition of a solution suitable for hyperbolicproblems; see Otto [Ott96] and [Vov02]. A notion of an entropy solution for degenerateparabolic problems with nonhomogeneous boundary conditions has also been definedby Mascia, Porretta, and Terracina in [MPT02]. It is interesting to notice that, intheir definition, they directly require that the entropy condition satisfy the entropycondition on the boundary (14) as stated in Proposition 4.1. We prove that thisproperty (14) is, in fact, a consequence of the entropy inequalities (4) and then followthe main lines of the uniqueness theorem proved in [MPT02].

3. Entropy process solution. The proof of the existence of a weak entropysolution to problem (1) lies in the study of the numerical solution uD defined by anFV method for problem (1) (see section 5.2). Theorem 5.1 states that the numericalsolution satisfies approximate entropy inequalities (see (50)), but the bounds on uD(a bound in L∞(Q) and a bound on the discrete H1-norm of ϕ(uD)) do not givestrong compactness, only weak compactness. Therefore, in order to be able to takethe limit of the nonlinear terms of uD (as Φ±

κ (uD), in particular), we have to turnto the notion of measure-valued solutions (see DiPerna [DiP85], Szepessy [Sze91]) or,equivalently, to the notion of entropy process solution defined by Eymard, Gallouet,and Herbin [EGH00]. In light of the following theorem, it appears that the notion ofentropy process solution is indeed well suited to compensate for the weakness of thecompactness estimates on the approximate solution uD and to deal with nonlinearexpressions of uD.

Theorem 3.1 (nonlinear convergence for the weak- topology). Let O be aBorel subset of R

m, R be positive, and (un) be a sequence of L∞(O) such that, forall n ∈ N, ||un||L∞ ≤ R. Then there exists a subsequence, still denoted by (un) andµ ∈ L∞(O × (0, 1)), such that

∀g ∈ C(R), g(un) −→∫ 1

0

g(µ(., α)) dα in L∞(O) weak- .

Now the notion of an entropy process solution can be defined.Definition 3.1 (weak entropy process solution). Let u be in L∞(Q × ( 0, 1)).

The function u is said to be an entropy process solution to problem (1) if

ϕ(u)− ϕ(u) ∈ L2(0, T ;H10 (Ω))(5)

and if u satisfies the following entropy inequalities for all κ ∈ [A,B], for all ψ ∈C∞c ([ 0, T )× R

d) such that ψ ≥ 0 and sgn±(ϕ(u)− ϕ(κ))ψ = 0 a.e. on Σ:∫Q

∫ 1

0

η±κ (u(t, x, α))ψt(t, x)

+(Φ±κ (t, x, u(t, x, α))−∇ (ϕ(u)(t, x)− ϕ(κ))

±)· ∇ψ(t, x)dαdxdt

+

∫Ω

η±κ (u0)ψ(0, x) dx +M

∫Σ

η±κ (u)ψ dγ(x)dt ≥ 0 .(6)

Notice that if the function u is an entropy process solution of problem (1), thenit satisfies condition (5), which means in particular that ϕ(u) does not depend on thelast variable α and is denoted by ϕ(u)(t, x).


Notation. We set Q = Q× (0, 1).We will now show that any entropy process solution actually reduces to an entropy

weak solution.

4. Uniqueness of the entropy process solution.Theorem 4.1 (uniqueness of the entropy process solution). Let u, v ∈ L∞(Q×

(0, 1)) be two entropy process solutions of problem (1) in accordance with Definition3.1. Suppose that Ω is either a polyhedral open subset of R

d or a strong C1,1 opensubset of R

d, and assume hypotheses (H1), (H2), (H3), (H4), and (H6) (or (H6Bis)).Then there exists a function w ∈ L∞(Q) such that

u(t, x, α) = w(t, x) = v(t, x, β) for almost every (t, x, α, β) ∈ Q× ( 0, 1)2 .Corollary 4.1 (uniqueness of the weak entropy solution). If Ω is either a

polyhedral open subset of Rd or a strong C1,1 open subset of R

d, and under hypotheses(H1), (H2), (H3), (H4), and (H6) (or (H6Bis)), problem (1) admits at most one weakentropy solution.

In the case where Ω is a polyhedral open subset of Rd, the proof of Theorem 4.1

is slightly more complicated than the proof in the case where Ω is a strong C1,1 opensubset of R

d. Besides, although the study of the FV method applied to (1) relieson Theorem 4.1 only in the case of Ω polyhedral, we wish to specify the validity ofTheorem 4.1 when Ω is C1,1. Indeed, problem (1) may of course be posed on suchan open set, and, in that case, Theorem 4.1 would be one of the major steps in theproof of the convergence of such an approximation, as for the vanishing viscosityapproximation, for example.

We therefore explain the proof of Theorem 4.1 in the case where Ω is C1,1 andthen indicate how to adapt it to the case where Ω is a polyhedral open subset of R

d

(see subsection 4.6).

4.1. Proof of Theorem 4.1: Definitions and notation.

4.1.1. Localization near the boundary. We suppose that Ω is a strong C1,1

open subset of Rd. In that case, there exists a finite open cover (Bν)0,... ,N of Ω and a

partition of unity (λν)0,... ,N on Ω subordinate to (Bν)0,... ,N such that, for ν ≥ 1, upto a change of coordinates represented by an orthogonal matrix Aν , the set Ω∩Bν isthe epigraph of a C1,1-function fν : R

d−1 → R; that is,

Ω ∩Bν = x ∈ Bν ; (Aν x)d > fν(Aν x) and

∂Ω ∩Bν = x ∈ Bν ; (Aν x)d = fν(Aν x) ,where y stands for (yi)1,d−1 if y ∈ R

d.Until the end of the proof of Theorem 4.1, the problem will be localized with the

help of a function λν . We drop the index ν and, for the sake of clarity, suppose thatthe change of coordinates is trivial: A = Id. We denote by Π = x, x ∈ Ω∩B ⊂ R

d−1

the projection of B ∩ Ω onto the (d − 1) first components, and Πλ = x, x ∈ supp(λ)∩Ω (see Figure 1). If a function ψ is defined on Σ, we denote by ψΣ the functiondefined on [0, T ) × B ∩ Q by ψΣ(t, x) = ψ(t, x, f(x)). Notice that the function ψΣ

does not depend on xd and that, by abusing the notation, we shall also denote by ψΣ

the restriction of ψΣ to [0, T )×Π. In the same way, if Li is defined on [0, T ]×Π, wealso denote by Li the function defined on [0, T )×B ∩Q by Li(t, x) = Li(t, x).

4.1.2. Weak notion of trace. An important step in the proof of the uniquenessof entropy process solutions is the derivation of the condition satisfied by any entropy


x

x

Ω_

= f ( x )

_x

B

λsupp( )

ΠΠλ

d

d

Fig. 1. Localization by λ in the ball B.

process solution on the boundary of the domain. This condition is the matter ofProposition 4.1. (It can be viewed as a kind of BLN condition [BLN79], balancedby second order terms issued from the degenerate parabolic part of the equation of(1).) In the course of the proof of Proposition 4.1, we need to define the normal traceof certain fluxes (Φ+

κ (t, x, u) − ∇(ϕ(u) − ϕ(κ))+, among others, for example) and,more precisely, to ensure the consistency of this definition of the normal trace withdifferent approximations. For that purpose, we turn to the work of Chen and Frid[CF02]. Adapted to our context, the main theorem of [CF02] is the following.

Theorem 4.2 (see Chen and Frid [CF02]). Recall that Q = (0, T )×Ω, and denoteby ν the outward unit normal to Q. Let F ∈ (L2(Q))d+1 be such that divF is a boundedRadon measure on Q. Then there exists a linear functional Tν on W 1/2,2(∂Q)∩C(∂Q)which represents the normal traces F · ν on ∂Q in the sense that, first, the followingGauss–Green formula holds: for all ψ ∈ C∞c (Q),

〈Tν , ψ〉 =∫Q

ψ divF +∫Q

∇ψ · F .(7)

Second, 〈Tν , ψ〉 depends only on ψ|∂Q, while, third, if (B, λ, f) is as above (subsectionlocalization near the boundary), then for all ψ ∈ C∞c ([0, T )× Ω),

〈Tν , ψλ〉 = − lims→0

1

s

∫ T

s

∫Π

∫ f(x)+s

f(x)

F · −∇f(x)

10

ψλdxd dx dt(8)

+

∫ s

0

∫Ω

F · 001

ψλdx dt

.

Let u be an entropy weak solution of problem (1). The entropy inequality (4)shows that the divergence of the field

F+κ (t, x) =

((u− κ)+

Φ+κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+

)is a bounded Radon measure on Q. This field belongs to (L2(Q))d+1, and accordingto the previous theorem, there exists a linear functional T +

ν,κ on W1/2,2(∂Q) ∩ C(∂Q)

which represents F+κ (t, x) · ν. Then, to define a notion of the normal trace of the flux


Φ+κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+, we set

〈T +n,κ, ψ〉 = 〈T +

ν,κ, ψ〉+∫

Ω

(u0 − κ)+ψ(0, x) dx ∀ψ ∈ C∞c ([0, T )× Ω) .(9)

This definition makes sense because the entropy weak solution assumes the values ofthe initial data u0:

lims→0

1

s

∫ s

0

∫Ω

(u− κ)+ψ dx dt =

∫Ω

(u0 − κ)+ψ(0, x) dx ,

as can be seen by choosing s−ts χ(0,s)(t)ψ as a test-function in (4). In particular,

〈T +n,κ, ψλ〉 depends only on ψ|Σ, and from (8) we can derive the formula

〈T +n,κ, ψλ〉

= − lims→0

1

s

∫ T

s

∫Π

∫ f(x)+s

f(x)

(Φ+κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+) ·

(−∇f(x)1

)ψλdxd dx dt

for all ψ ∈ C∞c ([0, T )× Ω).4.1.3. Mollifiers ρn and the cut-off function ωε. Technically, the heart of

the proof of uniqueness is the doubling of variables. This technique involves mollifiers,which are defined as ρn(t) = nρ(nt), where ρ is a nonnegative function of C∞c (−1, 0)such that

∫ 0

−1ρ(t) dt = 1. (Notice that the the support of the function ρ is located to

the left of zero.) For ε a positive number, ρε naturally denotes the map t → 1ερ(

tε ),

and we define Rn(t) =∫ −t

−∞ ρn(s) ds. Since the technique of doubling of variablesinterferes with a certain evaluation of the boundary behavior of the entropy processsolution (described by (14)), we need to define a cut-off function ωε built upon thesequence of mollifiers. We set

ωε(x) =

∫ 0

f(x)−xd

ρε(z) dz =

∫ 0

f(x)−xdε

ρ(z) dz.(10)

On Ω∩B, the function ωε vanishes in a neighborhood of ∂Ω and equals 1 if dist(x, ∂Ω) >ε; in particular, ωε → 1 in L1(Ω ∩B) and, if ψ ∈ H1(Ω), then∫

Ω

λψ · ∇ωε = −∫

Ω

div(λψ)ωεε→0→ −

∫Ω

div(λψ) = −∫∂Ω

λψ · n .

Roughly speaking, if F : Ω→ Rd, then −F · ∇ωε approaches the normal trace F · n.

To make this idea more precise, for the field F = Φ+κ (t, x, u) − ∇(ϕ(u) − ϕ(κ))+

we call upon the notion of normal trace defined above (subsection 4.1.2). Let ψ ∈C∞c ([0, T ) × Ω). Since ψ = ψ(1 − ωε) on Σ, 〈T +

n,κ, ψλ〉 = 〈T +n,κ, ψλ(1 − ωε)〉. The

definition of T +n,κ (see (9)) and the Gauss–Green formula (7) yield

〈T +n,κ, ψλ〉 =

∫Q

ψ(1−ωε)λ divF+κ +

∫Q

∇(ψ(1−ωε)λ)·F+κ +

∫Ω

(u0−κ)+ψ(1−ωε)λ dx.

Since 0 ≤ 1 − ωε ≤ 1 and ωε(x) → 1 for all x ∈ Ω ∩ B, the dominated convergencetheorem ensures that limε→0

∫Qψ(1− ωε)λ divF+

κ = 0 and

〈T +n,κ, ψλ〉 = − lim

ε→0

∫Q

[Φ+κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+] · ∇ωε ψλdx dt .


4.1.4. Otto entropy-fluxes. Let u ∈ L∞(Q × ( 0, 1)) be an entropy processsolution of problem (1) and κ ∈ [A,B]. Set Φ = Φ++Φ−. We denote by Gx(t, x, u, κ)the quantity

Gx(t, x, u, κ) = Φ(t, x, u(t, x, α), κ)−∇x|ϕ(u)(t, x)− ϕ(κ)| .(11)

For w ∈ R, the function Fϕ is defined by the formula

Fϕ(t, x, u, κ, w) = Gx(t, x, u, κ) + Gx(t, x, u, w)− Gx(t, x, κ, w) .(12)

4.2. A result of approximation.Lemma 4.1. Let U be a bounded open subset of R

q, q ≥ 1. If f ∈ L∞ ∩BV (U),then, given ε > 0, there exists g ∈ C(U) such that

g ≥ f a.e. on U and

∫U

(g(x)− f(x)) dx < ε .

This result may be false if f /∈ BV (U) (consider f = 11Q∩(0,1) on U = (0, 1)), butthis is not a necessary condition, because, on U = (0, 1), the function f = 11K , whereK is the triadic Cantor, can be approximated in L1(0, 1) by continuous functions gsuch that g ≥ f a.e. Indeed, we claim that, if E is a measurable subset of U , thenf = 11E satisfies the conclusion of Lemma 4.1 if and only if

m(E) = inf m(K) ; E ⊂ K , K compact .(13)

(Here, m denotes the Lebesgue measure on Rq.)

Before proving Lemma 4.1, let us justify this assertion. If (13) holds, then, givenε > 0, there exists a compact K of U such that E ⊂ K and m(K \ E) < ε. Sincethe Lebesgue measure is regular, there exists an open subset V of U such that K ⊂V ⊂ V ⊂ U and m(V \K) < ε. Then the function g : x → d(x,Rq \ V )/(d(x,K) +d(x,Rq \ V )) is continuous on R

q, g ≥ 11E , and∫U(g − 11E) < 2ε.

Conversely, suppose that, given ε > 0, there exists g ∈ C(U) such that g ≥ 11Eand

∫U(g − 11E) < ε. Then K = x ∈ U ; g(x) ≥ 1 is compact, E ⊂ K, and

m(K \ E) < ε.Proof of Lemma 4.1. Notice that, if E is a measurable subset of U such that

m(∂E) = 0, then (13) holds (consider the compact E). If E is a level set of a BVfunction, then E has almost surely a finite perimeter and, consequently, m(∂E) = 0,which ensures that 11E satisfies the conclusion of Lemma 4.1. This result may be seenas the heart of the proof. Indeed, first suppose that 0 ≤ f(x) ≤ 1 for every x ∈ U .For t ∈ [0, 1], set Et = x ∈ U ; f(x) < t. Then, for almost every t, Et is a setwith finite perimeter since f ∈ BV (U). Let (tn) be a sequence of reals dense in [0, 1]and such that t1 = 1; Etn is a set with finite perimeter for every n. We will define asequence of simple functions θn =

∑ni=1 α

ni 11An

iwhich approximate f from above and

such that each set Ani is built upon the level sets Eti . To that purpose, first define

θ1(x) = 1 for all x ∈ U . If n > 1, let k1, . . . , kn be an enumeration of 1, . . . , nsuch that tk1 > · · · > tkn . Set

Ani = Etki

\ Etki+1if 1 ≤ i < n ,

Ann = Etkn

and θn =∑n

i=1 tki 11Ani. Notice that (An

i )1≤i≤n is a partition of U and that Ani ⊂ Etki

;therefore, if x ∈ U , say x ∈ An

i , then θn(x) = tki> f(x) and θn ≥ f . Besides,


the sequence (Etki)1≤i≤n is decreasing, and this, together with the definition of A

ni ,

ensures that θn(x) ≤ ti if x ∈ Eti for 1 ≤ i ≤ n. Now, given x ∈ U and ε > 0,there exists n0 such that f(x) + ε > tn0

> f(x). Then, for every n ≥ n0, x ∈ Etn0

and, consequently, θn(x) ≤ tn0 < f(x) + ε. Thus, (θn) converges to f everywhere onU (in fact, the convergence is monotone, but we do not prove this fact), and, since0 ≤ θn ≤ 1, the dominated convergence theorem shows that

limn→+∞

∫U

θn − f = 0 .

However, for each fixed n, the function θn satisfies the conclusion of the lemma.Indeed, let ε > 0 be fixed. Since Etki+1

⊂ Etki, we have 11An

i= 11Etki

− 11Etki+1.

The functions 11Etkiand 11Etki+1

are in BV (U), by the definition of a set with finite

perimeter. Thus 11Aniis BV too, and An

i is a set with finite perimeter. As noticed

in the beginning of the proof, Ani satisfies (13), and there exists gi ∈ C(U) such that

gi ≥ tki 11Aniand

∫U(gi − tki

11Ani) < ε/n. Moreover, we can suppose that gi ≤ tki for

every i. Set g = max1≤i≤n gi. The function g is continuous on U , and g ≥ θn on Uby construction. It remains to compute ||g − θn||L1(U). If x ∈ An

i , then gi(x) = tki,

and the condition gj ≤ tkjenforces the maximum of the gj(x) to be reached for

j ∈ i, . . . , n. We then have

(g − θn)(x) = gj(x)− tki ≤ gj(x)− tkj 11Anj(x).

Indeed, if j = i, this is obvious, and if j > i, we have 11Anj(x) = 0, while tki ≥ 0.

Consequently, (g− θn)(x) ≤∑n

i=1(gj − tkj 11Anj)(x) and

∫U(g− θn) < n× ε/n = ε. If

n has been chosen such that∫U(θn − f) < ε, then g is relevant to the conclusion of

the lemma.We suppose that 0 ≤ f(x) ≤ 1 for every x ∈ U . For a general function f ∈

L∞∩BV (U), we can suppose, after an adequate modification of the function on a setof negligible measure, that −M ≤ f(x) ≤M for every x ∈ U , where M = ||f ||L∞(U).Then we consider the function f1 = (f + M)/(2M). Given ε > 0, there existsg1 ∈ C(U) such that g1(x) ≥ f1(x) and ||g1 − f1||L1(U) < ε/(2M) and g = 2Mg1 −Mis convenient.

4.3. Proof of Theorem 4.1 (preliminary): Boundary condition.Proposition 4.1 (boundary condition). Let u ∈ L∞(Q × ( 0, 1)) be an entropy

process solution of problem (1), and let Fϕ be defined by (12). Assume hypotheses(H1), (H2), (H3), (H4), and (H6) (or (H6Bis)). Then, for all κ ∈ [A,B], for allnonnegative ψ ∈ C∞c ([ 0, T )× R

d),

limε→0

∫QFϕ(t, x, u(t, x, α), κ, uΣ(t, x)) · ∇ωε(x)ψ(t, x)λ(x) dα dx dt ≤ 0 .(14)

In the case of a purely hyperbolic problem (ϕ′ = 0), inequality (14) is the bound-ary condition written by Otto [Ott96], equivalent to the BLN condition [BLN79] forBV solutions. If the problem is strictly parabolic (that is, ϕ′(u) ≥ Φmin > 0),then inequality (14) is trivially satisfied by any weak solution of the problem (1). In[MPT02], the condition (14) is listed among the conditions that an entropy solutionshould satisfy by definition. We refer to [MPT02] for a complete discussion of (14).


Proof of Proposition 4.1. We first aim to prove the following result: for everyκ ∈ [A,B], for every nonnegative ψ ∈ C∞c ([0, T )× R

d),

(15)

limε→0

∫Q[Φ+(t, x, u, κuΣ)−∇(ϕ(u)− ϕ(κuΣ))

+] · ∇ωε(x)ψ(t, x)λ(x) dα dx dt ≤ 0.

Fix κ ∈ [A,B]. In subsections 4.1.2 and 4.1.3, we defined a notion of normal trace forthe flux Φ+

κ (t, x, u)−∇(ϕ(u)−ϕ(κ))+ when u is an entropy weak solution of problem(1). Of course, the same can be done when u is an entropy process solution of problem(1); this time just consider the field F+

κ defined by

F+κ =

( ∫ 1

0(u− κ)+ dα∫ 1

0(Φ+

κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+)dα

).

Moreover, if T +n,κ still denotes the normal trace of the spatial part of F+

κ , for all

ψ ∈ C∞c ([ 0, T )× Rd),

〈T +n,κ, ψλ〉

(16)

=− lims→0

1

s

∫ T

s

∫ 1

0

∫Π

∫ f(x)+s

f(x)

(Φ+κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+) ·

(−∇f(x)1

)ψλdxd dx dt dα

and

〈T +n,κ, ψλ〉 = − lim

ε→0

∫Q[Φ+

κ (t, x, u)−∇(ϕ(u)− ϕ(κ))+] · ∇ωε ψλdx dt dα .(17)

Therefore, if ψ is a nonnegative function of C∞c ([ 0, T ) × Rd) such that sgn+(ϕ(u) −

ϕ(κ))ψ = 0 a.e. on (0, T )× ∂Ω, then, choosing ψ(1−ωε) as a test-function in (6), weget

−〈T +n,κ, ψλ〉 ≤M

∫Σ

(u− κ)+ ψ λdγ(x) dt .(18)

Now, we intend to define a notion of normal trace for the flux Φ+(t, x, u, uΣκ) −∇(ϕ(u)− ϕ(uΣκ))+. To that purpose, we set

F+=

( ∫ 1

0(u− uΣκ)+ dα∫ 1

0(Φ+(t, x, u, uΣκ)−∇(ϕ(u)− ϕ(uΣκ))+)dα

),(19)

and we prove the following lemma.Lemma 4.2. Let u ∈ L∞(Q × ( 0, 1)) be an entropy process solution of problem

(1), and let the field F+ ∈ (L2((0, T ) × B ∩ Q))d+1 be defined by (19). Assumehypotheses (H1), (H2), (H3), (H4), and (H6) (or (H6Bis)). Then, for every open

subset D of B such that D ⊂ B, the divergence of F+is a bounded Radon measure

on (0, T )×D ∩Q.Proof of Lemma 4.2. Set g = ∂t uΣ+divx F (t, x, uΣ)−∆ϕ(uΣ) . From hypothesis

(H6) we have g ∈ L1((0, T )×B ∩Q), and the function uΣ (which, we recall, belongsto W 1,1((0, T )× B ∩Q))) can be seen as an entropy solution of the equation ∂t w +


divx F (t, x, w) − ∆ϕ(w) = g with unknown w. The identity (uΣκ − κ)− = (uΣ −κ)− − (uΣ − κ⊥κ)− ensures that the function uΣκ satisfies the entropy inequality∫

Q

[(uΣκ− κ)− θt + [Φ

−(t, x, uΣκ, κ)−∇(ϕ(uΣκ)− ϕ(κ))−] · ∇θ ] dα dx dt+

∫Ω

(uΣκ(0, x)− κ)− θ(0) dx +

∫Qsgn−(uΣκ− κ) g θ dx dt dα ≥ 0

for every κ ∈ [A,B] and nonnegative function θ ∈ C∞c ([0, T ) × B ∩ Q). Now we usea result of comparison and assert that, for any nonnegative function θ ∈ C∞c ([0, T )×B ∩Q), we have∫

Q

[(u− uΣκ)+ θt + [Φ

+(t, x, u, uΣκ)−∇(ϕ(u)− ϕ(uΣκ))+] · ∇θ]dα dx dt

+

∫Ω

(u0 − uΣκ(0, x))+ θ(0) dx +

∫Qsgn+(u− uΣκ) g θ dx dt dα ≥ 0.(20)

This result of comparison, proved in [Car99] for entropy weak solution, remains truewhen applied to entropy process solutions. Notice that we state a result of comparisoninside [0, T )×Ω (the previous function θ vanishes on [0, T )×∂Ω); this point is crucial.A result of comparison on the whole domain Q is the object of Theorem 4.1, whichwe are actually proving. As a matter of fact, we would like to rule out the hypothesisthat θ vanishes on [0, T )× ∂Ω. Toward that end, first notice that (20) is still true ifθ ∈ C1

c ([0, T )× (B∩Ω)) and θ = 0 on [0, T )× (B∩∂Ω). Let θ ∈ C∞c ([0, T )× (B∩Ω)),define, for s > 0, hs(x) = min([xd − f(x)]/s, 1), and choose θ = θ hs in (20) to get

∫Q

[(u− uΣκ)+ θt + [Φ

+(t, x, u, uΣκ)−∇(ϕ(u)− ϕ(uΣκ))+] · ∇θ]hs dα dx dt

(21)

+

∫Ω

(u0 − uΣκ(0, x))+ θ(0)hs dx +

∫Qsgn+(u− uΣκ) g θ hs dx dt dα ≥ As +Bs,

where

As = −1s

∫ T

0

∫ 1

0

∫Π

∫ f(x)+s

f(x)

Φ+(t, x, u, uΣκ) · ∇x(xd − f(x)) θ dxd dx dt dα,

Bs =1

s

∫ T

0

∫Π

∫ f(x)+s

f(x)

∇(ϕ(u)− ϕ(uΣκ))+ · ∇x(xd − f(x)) θ dxd dx dt.

Let C be a bound of Φ+(t, x, z, w) · ∇x(xd− f(x)) in L∞(Q× [A,B]2). Such a boundexists and, for every s,

As ≥ −C T |Π| ||θ||L∞([0,T )×(B∩Ω)) .(22)

On the other hand, the term Bs can be decomposed as Bs = Bs +Bds , where

Bs = −1s

∫ T

0

∫Π

∫ f(x)+s

f(x)

∇x(ϕ(u)− ϕ(uΣκ))+ · ∇xf(x) θ dxd dx dt,

Bds =

1

s

∫ T

0

∫Π

∫ f(x)+s

f(x)

∂xd(ϕ(u)− ϕ(uΣκ))+ θ dxd dx dt.


Integration by parts with respect to x in Bs and integration by parts with respect toxd in B

ds (we use the fact that ϕ(u)(t, x, f(x)) = ϕ(uΣ)(t, x)) yields the following: for

almost every positive s (small enough),

Bs =1

s

∫ T

0

∫Π

∫ f(x)+s

f(x)

(ϕ(u)− ϕ(uΣκ))+divx∇xf(x) θ dxd dx dt,

Bds = −1

s

∫ T

0

∫Π

∫ f(x)+s

f(x)

(ϕ(u)− ϕ(uΣκ))+ ∂xdθ dxd dx dt

+1

s

∫ T

0

∫Π

(ϕ(u)− ϕ(uΣκ))+(t, x, f(x) + s) θ(x, f(x) + s)dx dt.

Notice that, first, the second term on the right-hand side of the previous equal-

ity in nonnegative; that, second, lims→0Bs = 0 and lims→01s

∫ T

0

∫Π

∫ f(x)+s

f(x)(ϕ(u) −

ϕ(uΣκ))+ ∂xdθ dxd dx dt = 0 (because the trace of ϕ(u) is ϕ(u)); and that, third,

hs converge to 1 in L1(B ∩ Ω). Consequently, letting s go to zero on both sides ofinequality (21) yields∫

Q

[(u− uΣκ)+ θt + [Φ

+(t, x, u, uΣκ)−∇(ϕ(u)− ϕ(uΣκ))+] · ∇θ]dα dx dt

+

∫Ω

(u0 − uΣκ(0, x))+ θ(0) dx +

∫Qsgn+(u− uΣκ) g θ dx dt dα ≥ lim inf

s→0As.

Let D be an open subset of B whose closure is a subset of B too. From (22), it appearsthat lim infs→0As can be viewed as the action of a certain distribution A∞ on θ andthat A∞ is a bounded Radon measure on [0, T )×D ∩Q. Since ∫ 1

0sgn+(u− uΣ) g dα

and (u0 − uΣκ(0, x))+δt=0 are bounded Radon measures on [0, T ) × D ∩ Q, the

previous inequality shows that the divergence of the field F+is a bounded Radon

measure on [0, T )×D ∩Q. This ends the proof of Lemma 4.2.Remark 4.1. If uΣ satisfies (H6Bis) instead of (H6), then uΣ can be seen as the

entropy solution of the equation ∂t w + divx F (t, x, w) − ∆ϕ(w) = g, with a sourceterm g which is a bounded Radon measure on (0, T ) × B ∩ Q. In the proof of theprevious lemma we used a theorem of comparison of Carrillo (Theorem 8 in [Car99])between two entropy solutions ui (i ∈ 1, 2) of the equation

∂tui + divF (t, x, ui)−∆ϕ(ui) = fi

(where fi ∈ L1) to derive the inequality (20). A careful study of the proof of the resultof comparison given by Carrillo shows that it still holds if f1 = 0 and f2 is a boundedRadon measure. Consequently, inequality (20) remains true under hypothesis (H6Bis)and Lemma 4.2 also.

As a consequence of this lemma, we can define a functional T +

n,κ, which representsthe normal trace of the flux Φ+(t, x, u, uΣκ)−∇(ϕ(u)−ϕ(uΣκ))+ on (0, T )×(∂Ω∩D) and satisfies the analogue of the relations (16) and (17), where κ has been replacedby uΣκ in these latter. (We use the fact that there exists an open set D such thatsupp(λ) ⊂ D ⊂ D ⊂ B to ensure that these limits make sense.)

Now, denote by S the set of all the functions v : (0, T )×Π→ R satisfying

v(t, x) =

Nv∑i=1

wi Li(t, x),(23)


where

∀i, wi ∈ R, Li ∈ C∞([0, T ]×Π), Li ≥ 0, andNv∑i=1

Li = 1 on [0, T ]×Πλ.(24)

We say that v ∈ S+if v ∈ S and admits a decomposition as (23) such that

wi ≥ uΣ a.e. on supp(Li) for all i. If v ∈ S and satisfies (23), we set

〈T +n,v κ, ψλ〉 =

Nv∑i=1

〈T +n,wi κ, Liψλ〉.

Notice that this is a notation and not a definition, because the decomposition (23)with wi, Li satisfying (24) is not unique. An immediate consequence of (18) is the

following: if v ∈ S+, then

−〈T +n,v κ, ψλ〉 ≤ 0 ∀ψ ∈ C∞c ([0, T )× R

d), ψ ≥ 0.(25)

Furthermore, we claim that, if v ∈ S+, then

〈T +n,v κ − T

+

n,κ, ψλ〉 ≤M√1 + ||∇xf ||2∞

Nv∑i=1

∫ T

0

∫Π

|wi − uΣ|ψλLi dx dt.(26)

Let us prove this result: from (16) we have 〈T +n,v κ−T

+

n,κ, ψλ〉 = − lims→0

∑Nv

i=1(Hi(s)+

Pi(s)), where

Hi(s) =1

s

∫ T

s

∫ 1

0

∫Π

∫ f(x)+s

f(x)

( Φ +(t, x, u, wiκ)

− Φ+(t, x, u, uΣκ)) ·( −∇f(x)

1

)Liψλdxd dx dt dα,

Pi(s) =1

s

∫ T

s

∫Π

∫ f(x)+s

f(x)

∇(( ϕ (u)− ϕ(uΣκ))+

− (ϕ(u)− ϕ(wiκ))+) ·( −∇f(x)

1

)Liψλdxd dx dt.

Since the function Φ+(t, x, u, v) isM -Lipschitz continuous with respect to v, uniformlywith respect to (t, x, u) ∈ Q× [A,B], we have

Hi(s) ≥ −1sM√1 + ||∇xf ||2∞

∫ T

0

∫Π

∫ f(x)+s

f(x)

|wiκ− uΣκ|Liψλdxd dx dt

≥ −1sM√1 + ||∇xf ||2∞

∫ T

0

∫Π

∫ f(x)+s

f(x)

|wi − uΣ|Liψλdxd dx dt .

Consequently,

Nv∑i=1

Hi(s) ≥ −1sM√1 + ||∇xf ||2∞

Nv∑i=1

∫ T

0

∫Π

∫ f(x)+s

f(x)

|wi − uΣ|ψλLi dxd dxdt,


and the limit of the right-hand side of this latter inequality can be explicitly computedsince the function ψλLi is smooth:

lims→0

Nv∑i=1

Hi(s) ≥ −M√1 + ||∇xf ||2∞

Nv∑i=1

∫ T

0

∫Π

|wi − uΣ|ψλLi dx dt .(27)

On the other hand, we have lim sups→0 Pi(s) ≥ 0. We will not detail the proof ofthis result, for it is identical to the justification of the fact that lim sups→0Bs ≥ 0 inthe proof of Lemma 4.2. Together with (27), the result lim sups→0 Pi(s) ≥ 0 yields(26). Furthermore, (26) combined with (25) shows that, if v ∈ S+

(v satisfies (23),with wi ≥ uΣ a.e. on supp(Li)), then

−〈T +

n,κ, ψλ〉 ≤M√1 + ||∇xf ||2∞

Nv∑i=1

∫ T

0

∫Π

(wi − uΣ)ψλLi dx dt.(28)

Since

〈T +

n,κ, ψλ〉 = − limε→0

∫Q[Φ+(t, x, u, uΣκ)−∇(ϕ(u)− ϕ(uΣκ))+] · ∇ωε ψλdx dt dα ,

our first aim, which is the proof of (15), will be reached if the right-hand side of(28) can be made as small as desired. Let us prove this fact: ε > 0. Since uΣ ∈L∞ ∩W 1,1((0, T ) × Π) (hypothesis (H6)), we have uΣ ∈ L∞ ∩ BV ((0, T ) × Π), andLemma 4.1 shows that there exists g ∈ C([0, T ]×Π) such that g ≥ uΣ a.e. on (0, T )×Πand

∫(0,T )×Π

g− uΣ < ε. Let η be a modulus of uniform continuity of g on [0, T ]×Π.The set (0, T )×Π (with compact closure) can be covered by a finite number of ballswith radius η centered in (0, T )×Π, say V1, . . . , VQ. Let (Li)1,Q be a regular partitionof unity subordinate to the open coverage (Vi) of [0, T ]×Π. For a certain (ti, xi) ∈ Vi,set wi = g(ti, xi) + ε and define v =

∑Qi=1 wi Li. Then v ∈ S+

and

Q∑i=1

∫ T

0

∫Π

(wi − uΣ)ψλLi dx dt=

∫ T

0

∫Π

(v − uΣ)ψλdx dt

=

∫ T

0

∫Π

(v − g)ψ λdx dt+

∫ T

0

∫Π

(g − uσ)ψ λdx dt

≤ 2||ψ λ||∞ T |Π| ε.

This completes the proof of (15). Similarly, we can prove

(29)

limε→0

∫Q[Φ−(t, x, u, κ⊥uΣ)−∇(ϕ(u)− ϕ(κ⊥uΣ))

−] · ∇ωε(x)ψ(t, x)λ(x) dα dx dt ≤ 0

for every κ ∈ [A,B] and for every nonnegative ψ ∈ C∞c ([0, T )×Rd). Then Proposition

4.1 follows from the formula

Fϕ(t, x, u, κ, w)=[Φ +(t, x, u, κw)−∇(ϕ(u)− ϕ(κw))+]+[Φ−(t, x, u, κ⊥w)−∇(ϕ(u)− ϕ(κ⊥w))−] .


4.4. Proof of Theorem 4.1 (step 1): Inner comparison. Let u and v ∈L∞(Q× ( 0, 1)) be two entropy process solutions of problem (1). The following resultof comparison between u and v involving test-functions which vanish on the boundaryof Ω can be proved (see [Car99] or [EGHM02]).

Proposition 4.2 (inner comparison). Let u and v ∈ L∞(Q × ( 0, 1)) be twoentropy process solutions of problem (1). Assume hypotheses (H1), (H2), (H3), and(H4). Let ζ be a nonnegative function of C∞([ 0, T )× R

d × [ 0, T )× Rd) such that

∀(s, y) ∈ Q , (t, x) −→ ζ(t, x, s, y) ∈ C∞c ([ 0, T )× Ω),∀(t, x) ∈ Q , (s, y) −→ ζ(t, x, s, y) ∈ C∞c ([ 0, T )× Ω).

Then we have

∫Q

∫Q

|u(t, x, α)− v(s, y, β)|(ζt + ζs)

+Gx(t, x, u(t, x, α), v(s, y, β)) · ∇xζ+Gy(s, y, v(s, y, β), u(t, x, α)) · ∇yζ−∇x|ϕ(u)(t, x)− ϕ(v)(s, y)| · ∇yζ−∇y|ϕ(u)(t, x)− ϕ(v)(s, y)| · ∇xζ

dαdxdtdβdyds

+

∫Q

∫Ω

|u0(x)− v(s, y, β)| ζ(0, x, s, y) dx dβ dy ds

+

∫Q

∫Ω

|u0(y)− u(t, x, α)| ζ(t, x, 0, y) dy dα dx dt ≥ 0.

(30)

4.5. Proof of Theorem 4.1 (step 2): General test-function. We now followthe lines of the proof of uniqueness given by Mascia, Porretta, and Terracina in[MPT02].

First, we would like to consider test-functions which do not necessarily vanish on∂Ω and are localized into the ball B. For x ∈ R

d−1, set ρm(x) = ρm(x1) · · · ρm(xd−1)and define the function ξ by

ξ(t, s, x, y) = ψ(t, x) ρl(t− s) ρm(x− y) ρn(xd − yd) .(31)

We took care to choose ρ satisfying supp(ρ) ⊂ [−1, 0) to ensure

∀(t, x) ∈ Q , (s, y) −→ ξ(t, s, x, y) ∈ C∞c (Q) ,∀(t, s, x) ∈ [ 0, T )× [ 0, T )× supp (λ), suppy ξ(t, s, x, ·) ⊂ B.

(32)

For ε > 0 define ζ to be the function

ζ : (t, s, x, y) −→ ωε(x) ξ(t, s, x, y)λ(x).

Then, for m large enough compared with n, the assumptions of Proposition 4.2 aresatisfied, and, with this particular choice of function ζ, inequality (30) turns into theinequality


∫Q

∫Q

|u− v|ωε(x) ((ξλ)t + (ξλ)s)+(Gx(t, x, u, v) · ∇x(ξ λ)

+Gy(t, y, v, u) · ∇y(ξ λ))ωε(x)

−(∇x|ϕ(u)− ϕ(v)| · ∇y(ξ λ)

+∇y|ϕ(u)− ϕ(v)| · ∇x(ξ λ))ωε(x)

dxdtdαdydsdβ

+

∫Q

∫QGx(t, x, u, v) · ∇ωε(x) ξ λ dα dx dt dβ dy ds

−∫Q

∫Q∇y|ϕ(u)− ϕ(v)| · ∇ωε(x) ξ λ dx dt dα dy ds dβ

+

∫Ω

∫Q|u0(x)− v| (ξ λ)(0, x, y)ωε(x) dxδβ dy ds ≥ 0,

where

u = u(t, x, α) and v = v(s, y, β).

Using formula (12), this inequality can be rewritten as

∫Q

∫Q

|u− v|ωε(x) ((ξ λ)t + (ξλ)s)

+ (Gx(t, x, u, v) · ∇x(ξ λ) + Gy(t, y, v, u) · ∇y(ξ λ))ωε(x)− (∇x|ϕ(u)− ϕ(v)| · ∇y(ξ λ)+∇y|ϕ(u)− ϕ(v)| · ∇x(ξ λ))ωε(x)

dαdxdtdβdyds

+

∫Q

∫QFϕ(t, x, u, v, uΣ) · ∇ωε(x) ξ λ dα dx dt dβ dy ds

+

∫Ω

∫Q|u0(x)− v| (ξ λ)(0, x, y)ωε(x) dx dy ds dβ ≥ A+B + C,

(33)

where

A =

∫Q

∫Q

∇y|ϕ(u)− ϕ(v)| · ∇ωε(x) ξ λ dx dt dy ds,

B = −∫Q

∫QGx(t, x, v, uΣ) · ∇ωε(x) ξ λ dα dx dt dβ dy ds,

C =

∫Q

∫QGx(t, x, u, uΣ) · ∇ωε(x) ξ λ dα dx dt dβ dy ds.

Using Proposition 4.1 and taking the limit of both sides of the previous inequalitywith respect to ε then yields

∫Q

∫Q

|u− v| ((ξ λ)t + (ξ λ)s)+Gx(t, x, u, v) · ∇x(ξ λ) + Gy(t, y, v, u) · ∇y(ξ λ)

−∇x|ϕ(u)− ϕ(v)| · ∇y(ξ λ) +∇y|ϕ(u)− ϕ(v)| · ∇x(ξ λ)

dα dx dt dβ dy ds+

∫Ω

∫Q|u0(x)− v| (ξ λ)(0, x, y) dβ dy ds dx ≥ lim

ε→0(A+B + C ),


or (using formula (11))

∫Q

∫Q

|u− v| ((ξ λ)t + (ξ λ)s)+Φ(t, x, u, v) · ∇x(ξ λ) + Φ(t, y, v, u) · ∇y(ξ λ)

− (∇x|ϕ(u)− ϕ(v)|+∇y|ϕ(u)− ϕ(v)|) · (∇y +∇x)(ξ λ)

dα dx dt dβ dy ds+

∫Ω

∫Q|u0(x)− v| (ξ λ)(0, x, y) dβ dy ds dx ≥ lim

ε→0(A+B + C ).

(34)

Now, we intend to pass to the limit on l, m, and n in the previous inequality.We will do so (on l and m and, eventually, on n), but notice that the study of thebehavior of A, B, and C as [ε→ 0] and the doubling variable technique itself interferewith each other.

Using the definition of ξ from (31), it appears that C does not depend on l, m,and n:

C =

∫QGx(t, x, u, uΣ) · ∇ωε(x)ψ λdα dx dt.

Moreover, inequality (34) can be rewritten as

∫Q

∫Q

|u− v| ρl ρm ρn(ψ λ)t+Φ(t, x, u, v) · ∇x(ψ λ)ρl ρm ρn

− (∇x|ϕ(u)− ϕ(v)|+∇y|ϕ(u)− ϕ(v)|) · ∇x(ψ λ) ρl ρm ρn

dα dx dt dβ dy ds

+

∫Ω

∫Ω

|u0(x)− u0(y)| (ψ λ)(0, x) ρm ρn dx dy ≥ limε→0

(A+B + C ) +D + E,

(35)

where

D = −∫Q

∫Q[Φ(t, x, u, v)− Φ(t, y, u, v)] · ∇x(ρl ρm ρn)ψ λdα dx dt dβ dy ds,

E =

∫Ω

∫Q|u0(y)− v| (ψ λ)(0, x) ρl(−s) ρm ρn dβ dy ds dx.

The term E can be estimated by using the fact that the solution v completely satisfies

the initial condition, which means, for example, that ess lims→0+

∫Ω

∫ 1

0|v(s, y, α) −

u0(y)| dβ dy = 0. On the other hand, if the flux function F does not depend onthe (t, x)-variables, then D = 0, and more generally, one can prove (see [CH99])D + E ≥ H, where

H = −C(F,ψ) sup ∫

Q|v(s, y, yd, β)− v(s+ σ, y + h, yd + k, β)|ds dy dyd dβ ;

|σ| ≤ 1l, |h| ≤ 1

m, |k| ≤ 1

n

.(36)

Notice that, by continuity of the translations in L1, we have liml,m,n→+∞H = 0.


4.5.1. Study of A + B. Going back to the study of A, B, we write A + B =I + Jy + Jx, where

I = −∫Q

∫Q(Φ(t, x, v, uΣ(t, x)) · ∇ωε(x) ξ λ dα dx dt dβ dy ds,

Jy =

∫Q

∫Q

∇y|ϕ(u)(t, x)− ϕ(v)(s, y)| · ∇ωε(x) ξ λ dx dt dy ds,

Jx =

∫Q

∫Q

∇x|ϕ(v)− ϕ(uΣ(t, x))| · ∇ωε(x) ξ λ dx dt dy ds.

Recall that

∇ωε(x) = ρε(f(x)− xd)

(−∇f(x)1

),

so that

I = limε→0

I

= −∫Q

∫[0,T )×Π×(0,1)

(Φ(t, x, f(x), v, uΣ(t, x)) ·(−∇f(x)

1

)(ξ λ)Σx

dαdxdtdβdyds,

where the index Σx indicates that the transformation concerns only the x variable.Here, for example, (ξ λ)Σx

(t, x, y) = ξ(t, x, f(x), y)λ(x, f(x)). To study the term Jx,we notice that the function uΣ does not depend on xd, and thus

Jx = limε→0

Jx = −∫

[0,T )×Π

∫Q

∇x |ϕ(v)− ϕ(uΣ(t, x))| · ∇f(x) (ξ λ)Σxdx dt dy ds.

Integration by parts with respect to x in Jx yields Jx = Jxf + Jx

ψ + Jxρm+ Jx

ρn, where

Jxf =

∫[0,T )×Π

∫Q

|ϕ(v)− ϕ(uΣ)|∆f(x) (ψ λ)Σxρl(t− s)

×ρm(x− y) ρn(f(x)− yd) dx dt dy ds,

Jxψ =

∫[0,T )×Π

∫Q

|ϕ(v)− ϕ(uΣ)|∇f(x)·∇x ((ψ λ)Σx

)ρl(t− s) ρm(x− y)ρn(f(x)− yd)dxdtdyds,

Jxρm=

∫[0,T )×Π

∫Q

|ϕ(v)− ϕ(uΣ)|∇f(x)·∇x ρm(x− y) ρn(f(x)− yd) ρl(t− s)ψ λdx dt dy ds,

Jxρn=

∫[0,T )×Π

∫Q

|ϕ(v)− ϕ(uΣ)||∇f(x)|2ρl(t− s)

×ρm(x− y) ρ′n(f(x)− yd) (ψ λ)Σxdxdtdyds.

On the other hand, via integration by parts in Jy with respect to y, and recalling thatthe boundary condition ϕ(u) = ϕ(u) on Σ is strongly satisfied according to Definition3.1, we get

Jy = limε→0

Jy

= −∫

[0,T )×Π

∫Q

|ϕ(uΣ(t, x))− ϕ(v)|(−∇f(x)

1

)· ∇y(ξ λ)(t, s, x, f(x), y) dx dt dy ds,


and, developing the scalar product,

Jy =

∫[0,T )×Π

∫Q

|ϕ(uΣ(t, x))− ϕ(v)|∇f(x) · ∇y(ξ λ)(t, s, x, f(x), y)dy dx dt ds

−∫

[0,T )×Π

∫Q

|ϕ(uΣ(t, x))− ϕ(v)|∂yd (ξ λ)(t, s, x, f(x), y) dy dx dt ds

= −Jxρm+

∫[0,T )×Π

∫Q

|ϕ(uΣ)− ϕ(v)|ρl(t− s)

× ρm(x− y) ρ′n(f(x)− yd) (ψ λ)Σxdy dx dt ds,

so that

Jx + Jy = Jxf + Jx

ψ +

∫[0,T )×Π

∫Ω

|ϕ(uΣ)− ϕ(v)| (1 + |∇f(x)|2)

× ρl(t− s) ρm(x− y)ρ′n(f(x)− yd) (ψ λ)Σxdx dt dy ds.

In particular, no derivatives of the functions ρm or ρl appear in Jx + Jy. Hence,

summing up by v the quantity v(t, x, yd, β) and passing to the limit [l,m → +∞] inlimε→0(A+B) = I + Jx + Jy, we get

liml,m→+∞

limε→0

(A+B) = I + Jf + Jψ + Jρn ,

with

I = −∫

[0,T )×Π×(0,1)

∫ ∞

0

∫ 1

0

Φ(t, x, f(x), v, uΣ)

·(−∇f(x)

1

)ρn(f(x)− yd)(ψ λ)Σxdxdtdαdyddβ,

Jf =

∫[0,T )×Π

∫ ∞

0

|ϕ(v)− ϕ(uΣ)|∆f(x) (ψ λ)Σxρn(f(x)− yd) dx dt dyd,

Jψ =

∫[0,T )×Π

∫ ∞

0

|ϕ(v)− ϕ(uΣ)|∇f(x) · ∇x ((ψ λ)Σx) ρn(f(x)− yd) dx dt dyd,

Jρn =

∫[0,T )×Π

∫ ∞

0

|ϕ(v)− ϕ(uΣ)| (1 + |∇f(x)|2) ρ′n(f(x)− yd) (ψ λ)Σx dx dt dyd.

To compute the limit as n tends to +∞ of the four preceding terms, first recall thattrace((ϕ(v))− ϕ(uΣ)) = 0, and that, consequently,

limn→+∞ Jf = 0 and lim

n→+∞ Jψ = 0.

Besides, we note that

∆ω1/n(x) = −ρ′n(f(x)− xd) (1 + |∇f(x)|2) + ρn(f(x)− xd)∆f(x),

so that, replacing yd by xd in Jρn, we have

Jρn = −∫Q

|ϕ(v)− ϕ(uΣ(t, x))|∆ω1/n(x) (ψ λ)(t, x, f(x)) dx dt+ Jf

=

∫Q

∇|ϕ(v)− ϕ(uΣ(t, x))| ∇ω1/n(x) (ψ λ)(t, x, f(x)) dx dt+ ε1n .


Here, the quantity ε1n = Jf +∫Q|ϕ(v)− ϕ(uΣ(t, x))| ∇ω1/n(x) · ∇(ψ λ)Σx

dx dt tendsto zero when n→ +∞. Moreover,

I = −∫QΦ(t, x, v, uΣ) · ∇ω1/n(x) (ψ λ)Σx

dβ dx dt + ε2n ,

where ε2n =∫Q (Φ(t, x, v, uΣ) − Φ(t, x, f(x), v, uΣ)) · ∇ω1/n(x) (ψ λ)Σx

dβ dx dt tendsto zero when n→ +∞.

Using formula (11), we get

lim infn→+∞ lim

l,m→+∞limε→0

(A+B)

= − lim supn→+∞

∫QGx(t, x, v(t, x, β), uΣ) · ∇ω1/n(x) (ψ λ)Σ dx dt dβ .

Starting from inequality (35) and taking the limit with respect to l, m, then thelimit with respect to n of both sides yields

(37)

∫Q

∫ 1

0

∫ 1

0

[|u− v| (ψλ)t + Gx(t, x, u, v) · ∇(ψλ) ] dβ dα dx dt

≥

− lim

n→+∞

∫Q

∫ 1

0

Gx(t, x, v(t, x, β), uΣ(t, x)) · ∇ω1/n (ψ λ)(t, x, f(x)) dβ dx dt

+ limε→0

∫Q

∫ 1

0

Gx(t, x, u, uΣ(t, x)) · ∇ωε(x) (ψ λ)(t, x, f(x)) dα dx dt+ lim

n→+∞ liml,m→+∞

H

.

Since limn→+∞ liml,m→+∞H = 0 (see (36)), the right-hand side of (37) is an anti-symmetric function in (u, v), while the left-hand side of (37) is a symmetric functionof (u, v). We therefore have∫

Q

∫ 1

0

[|u− v| (ψλ)t + Gx(t, x, u, v) · ∇(ψλ) ] dβ dα dx dt ≥ 0.(38)

Now, recall that λ = λα is an element of the partition of unity (λα)0≤α≤N ; summingthe previous inequality over α ∈ 0, . . . , N yields∫

Q

∫ 1

0

[|u− v|ψt + Gx(t, x, u, v) · ∇ψ ] dβ dα dx dt ≥ 0.(39)

We define the nonnegative function ψ0 by ψ0(t, x) = ψ0(t) = (T − t)χ(0,T )(t), andapply (39) with ψ0 as a test-function to get∫ T

0

∫Ω

∫ 1

0

∫ 1

0

|u(t, x, α)− v(t, x, β)| dβ dα dx dt ≤ 0.

Consequently, we have u(t, x, α) = v(t, x, β) for a.e. (t, x, α, β) ∈ Q × ( 0, 1) × ( 0, 1).Defining the function w by the formula

w(t, x) =

∫ 1

0

u(t, x, α) dα

and accounting for the product structure of the measurable space Q× ( 0, 1)× ( 0, 1),we conclude

u(t, x, α) = w(t, x) = v(t, x, β) for a.e. (t, x, α, β) ∈ Q× ( 0, 1)2.


4.6. Proof of Theorem 4.1 for Ω a bounded polyhedral subset. Let d bethe Euclidean distance on R

d. Denote by (∂Ωi)i=1,...,N the faces of Ω, and by ni theoutward unit normal to Ω along ∂Ωi. For ε > 0 small, let B

εi be the subset of all x ∈ Ω

such that d(x, ∂Ωi) < ε and d(x, ∂Ωi) < d(x, ∂Ωj) if i = j; define Gεi to be the largest

cylinder generated by ni included in Bεi , and set ∆

εi = Bε

i \Gεi , Ωε = Ω \ (∪1,N∆

εi ),

and bε = 11Ωε/2 ρε/4. We have meas(Ω \ Ωε) ≤ Cε2. If λi ∈ C∞c (Rd) is such that

supp(λi) ∩ ∂Ω ⊂ ∂Ωi and such that the orthogonal projection of supp(λi) on theaffine hyperplane determined by ∂Ωi is included in ∂Ωi, then of course the wholeprevious proof explained in the case where Ω is C1,1 applies here (we look at a half-space), to give a result of comparison on supp(λ). Otherwise, for such a choice offunction λi, (38) is true. Equation (38) is also still true if λ = λ0, where λ0 ∈ C∞c (Rd)and supp(λ0) ⊂ Ω (use Proposition 4.2). Since the function bε can be written asbε =

∑i=0,N λi for functions λi as above, we have∫

Q

∫ 1

0

[|u− v| (ψbε)t + Gx(t, x, u, v) · ∇(ψbε) ] dβ dα dx dt ≥ 0.(40)

Equation (40) can be rewritten as∫Q

∫ 1

0

[|u− v|ψt + Gx(t, x, u, v) · ∇ψ] dβ dα dx dt ≥ αε,

where αε =∫Q∫ 1

0Gx(t, x, u, v) · ∇bεψ dβ dα dx dt tends to zero when ε → 0. Indeed,

we have ∇bε = 0 on Ωε, so that, setting Rε = (0, T )× (Ω \ Ωε)× (0, 1)2, we have

αε≤ ||ψ||L∞ ||Gx(t, x, u, v)||L1(Rε)||∇bε||L∞(Rε)

≤ ||ψ||L∞meas(Rε)1/2||Gx(t, x, u, v)||L2(Rε)||11Ωε/2

||L∞(Rε)||∇ρε/4||L1(Rε)

≤C(T, ψ) ε · ||Gx(t, x, u, v)||L2(Rε) · 1ε ,

and we conclude by using ||Gx(t, x, u, v)||L2(Rε) → 0 when ε → 0. We thus obtain(39), from which Theorem 4.1 follows.

5. The FV scheme. The mesh used to discretize problem (1) has to be regularenough to ensure the consistency of the numerical fluxes, mainly because a secondorder problem is considered (at least when the function ϕ is not constant). This isspecified in the following section.

5.1. Assumptions and notation. We set d to be the Euclidean distance onRd and denote by γ the (d− 1)-Hausdorff measure on ∂Ω.Definition 5.1 (admissible mesh of Ω). An admissible mesh of Ω consists of a

set T of open bounded polyhedral convex subsets of Ω called control volumes, a familyE of subsets of Ω contained in hyperplanes of R

d with positive measure, and a familyof points (the “centers” of control volumes) satisfying the following properties:

(i) The closure of the union of all control volumes is Ω.(ii) For any K ∈ T , there exists a subset EK of E such that ∂K = K\K = ∪σ∈EK

σ.Furthermore, E = ∪K∈T EK .

(iii) For any (K,L) ∈ T 2 with K = L, either the “length” (i.e., the (d − 1)-dimensional Lebesgue measure) of K ∩ L is 0 or K ∩ L = σ for some σ ∈ E. In thelatter case, we shall write σ = K|L and Eint = σ ∈ E ,∃(K,L) ∈ T 2, σ = K|L. Forany K ∈ T , we shall denote by NK the set of neighbor control volumes of K, i.e.,NK = L ∈ T ,K|L ∈ EK.


(iv) The family of points (xK)K∈T is such that xK ∈ K (for all K ∈ T ), and, ifσ = K|L, it is assumed that the straight line (xK , xL) is orthogonal to σ.

Given a control volume K ∈ T , we will denote by m(K) its measure and by Eext,Kthe subset of the edges of K included in the boundary ∂Ω. If L ∈ NK , m(K|L) willdenote the measure of the edge between K and L, and TK|L the “transmissibility”

through K|L, defined by TK|L =m(K|L)d(xK ,xL) . Similarly, if σ ∈ Eext,K , we will denote

by m(σ) its measure and by τσ the “transmissibility” through σ, defined by τσ =m(σ)

d(xK ,σ) . One also denotes by Eext the union of the edges included in the boundary ofΩ: ∪K∈T Eext,K . The size of the mesh T is defined by

size(T ) = maxK∈T

diam(K),

and we introduce the following geometrical factor, linked with the regularity of themesh, defined by

reg(T ) = minK∈T ,σ∈EK

d(xK , σ)

diam(K).

Remark 5.1. Some examples of meshes satisfying these assumptions are thetriangular meshes, which verify the acute angle condition (in fact this condition maybe weakened to the Delaunay condition), the rectangular meshes, or the Voronoımeshes; see [EGH99] or [EGH00] for more details.

Definition 5.2 (time discretization of (0, T )). A time discretization of (0, T ) isgiven by an integer value N and by an increasing sequence of real values (tn)n∈[[0,N+1]]

with t0 = 0 and tN+1 = T . The time steps are then defined by δtn = tn+1 − tn, forn ∈ [[0, N ]].

Definition 5.3 (space-time discretization of Q). A finite volume discretizationD of Q is a family D = (T , E , (xK)K∈T , N ,(tn)n∈[[0,N ]]), where T , E, (xK)K∈T isan admissible mesh of Ω according to Definition 5.1 and N , (tn)n∈[[0,N+1]] is a timediscretization of (0, T ) according to Definition 5.2. For a given FV discretization D,one defines

size(D) = max(size(T ), (δtn)n∈[[0,N ]]) and reg(D) = reg(T ).5.2. The FV scheme. We may now define the FV discretization of (1). Let D

be a FV discretization of Q according to Definition 5.3. First, the initial and boundarydata are discretized by setting

U0K =

1

m(K)

∫K

u0(x)dx ∀K ∈ T(41)

and

Un+1σ =

1

δtn m(σ)

∫ tn+1

tn

∫σ

u(t, x)dγ(x)dt ∀σ ∈ Eext,∀n ∈ [[0, N ]].(42)

An implicit FV scheme for the discretization of problem (1) is given by the fol-lowing set of nonlinear equations with unknowns UD = (Un+1

K )K∈T ,n∈[[0,N ]]: ∀K ∈T ,∀n ∈ [[0, N ]],(43)

Un+1K − Un

K

δtnm(K) +

∑σ∈EK

m(σ)Fn+1K,σ (U

n+1K , Un+1

Kσ)−

∑σ∈EK

τσ(ϕ(Un+1Kσ

)− ϕ(Un+1K )) = 0,


where

Un+1Kσ

=

Un+1L if σ = K|L,

Un+1σ if σ ∈ Eext,(44)

and where the function Fn+1K,σ is a monotonous flux consistent with the function F ,

which means that• for all v ∈ R, u → Fn+1

K,σ (u, v) is a nondecreasing function and for all u ∈ R,

v → Fn+1K,σ (u, v) is a nonincreasing function,

• Fn+1K,σ (u, v) = −Fn+1

K,σ (v, u) for all (u, v) ∈ R2,

• Fn+1K,σ is M -Lipschitz continuous with respect to each variable,

• Fn+1K,σ (s, s) =

1δtn

1m(σ)

∫ tn+1

tn

∫σF (x, t, s) · nK,σdγ(x) dt.

The Godunov scheme and the splitting flux scheme of Osher may be the mostcommon examples of schemes with monotone fluxes.

We call an approximate solution the piecewise constant function uD defined a.e.on Q by

uD(t, x) = Un+1K , t ∈ (tn, tn+1), x ∈ K.(45)

5.3. Monotony of the scheme and direct consequences. As already said inthe introduction, it is a necessity to select a physically admissible solution by means ofthe entropy inequalities. The schemes with monotonous fluxes are well known to addnumerical viscosity to the equations. They are L∞ stable, and they are monotonousso that they respect discrete entropy inequalities. In other words, continuous entropyinequalities have their discrete analogue, and they are respected by any solution of(41)–(44). This is summarized in the following proposition.

Proposition 5.1 (monotony). Assume hypotheses (H1), (H2), (H3), and (H4).Then there exists a unique solution to the scheme. Moreover, this solution satisfies thefollowing maximum principle and discrete entropy inequalities: ∀K ∈ T ,∀n ∈ [[0, N ]],

A ≤ Un+1K ≤ B,(46)

η±κ (Un+1K )− η±κ (U

nK)

δtnm(K) +

∑σ∈EK

m(σ)Φ±,n+1K,σ,κ (U

n+1K , Un+1

Kσ)(47)

−∑σ∈EK

τσ(η±κ (ϕ(U

n+1Kσ

))−η±κ (ϕ(Un+1K ))

) ≤ 0,where Φ+,n+1

K,σ,κ and Φ−,n+1K,σ,κ are the numerical entropy-fluxes defined by

Φ+,n+1K,σ,κ (u, v) = Fn+1

K,σ (uκ, vκ)− Fn+1K,σ (κ, κ) and(48)

Φ−,n+1K,σ,κ (u, v) = Fn+1

K,σ (κ, κ)− Fn+1K,σ (u⊥κ, v⊥κ).

Proof. We give only some elements of the proof of this proposition because itconsists of rewriting the proofs of three lemmas that can be found in [EGHM02](Lemmas 3.1, 3.3, and 3.4 there) in the case where the convective flux q(x, t)f(u) isreplaced by a more general flux F (x, t, u) and the Kruzhkov entropies are replaced bythe semi-Kruzhkov entropies as in the work of Vovelle [Vov02].

We follow the classical framework of implicit FV schemes for conservation laws(see [EGH00]). The function UD is defined in an implicit way, so we first show, usingthe monotony of the scheme, that if a function UD is a solution to the scheme, then


it satisfies the discrete inequalities (47). Then we derive the maximum principle (46)that provides a result of existence by use of the Leray–Schauder theorem. Uniquenessof UD is proved by using a method analogous to the one used to prove the discreteentropy inequalities.

5.4. A priori estimates. The inequalities derived from the properties of monot-ony and local conservation are L∞ and L1 estimates. We will prove now L2 estimates.We introduce a discretization UD = (U

n+1K )K∈T ,n∈[[0,N ]] of u defined by

Un+1K =

1

δtn

1

m(K)

∫ tn+1

tn

∫K

u dx dt ∀K ∈ T , ∀n ∈ [[0, N ]] .

Proposition 5.2 (L2(0, T,H1(Ω)) and weak BV estimate). Assume hypotheses(H1), (H2), (H3), (H4), and (H5). Let uD be the approximate solution defined by(41)–(44), and assume that reg(D) ≥ ξ, where ξ > 0. Then there exists a constant Cdepending only on ξ, T , Ω, Lip(ϕ), M , u, A, B such that

(ND(ζ(uD)))2=

N∑n=0

δtn∑K∈T

12

∑σ∈Eint,K

τσ(ζ(Un+1K )− ζ(Un+1

Kσ))2

+∑

σ∈Eext,K

τσ(ζ(Un+1K )− ζ(Un+1

Kσ))2

≤ C

and

N∑n=0

δtn∑K∈T

1

2

∑σ∈Eint,K

m(σ) maxUn+1

K≤c≤d≤Un+1

Kσ

((Fn+1

K,σ (d, c)− Fn+1K,σ (d, d))

2

+ (Fn+1K,σ (d, c)− Fn+1

K,σ (c, c))2) ≤ C.(49)

Remark 5.2. The inequality (49) is called the “weak BV inequality.” See [EGH00],[CGH93], or [CH99].

Proof. As for Proposition 5.1, the proof has already been done in a simpler case in[EGHM02] (Proposition 3.1). The details of the proof differ only by some argumentsthat can be found in [Vov02].

These estimates are discrete energy estimates. They are obtained by multiplying(41)–(44) by δtn(U

n+1K − Un+1

K ) and summing over K ∈ T and n ∈ [[0, N ]]. In theproof, we separate terms that contain only UD from terms containing UD and UD.Then we use the Cauchy–Schwarz inequality and regularity hypotheses (H5) on u tocontrol the second type of terms. To get a bound on ND(UD), which is a discreteL2(0, T,H1)-norm for UD, we use the following inequality proved in [EGH99]:

ND(u) ≤ C(reg(D))‖∇u‖L2(Q).

This is a consequence of the local conservativity of the scheme combined with theconsistency of the numerical fluxes.

The last ingredient is the assumption divx(F (x, t, u)) = 0, which ensures thatthe boundary terms in the discrete integrations-by-parts concerning the hyperbolicterms can be controlled. The constant C depends on ξ, m(Ω), T , B, A, Lip(Fn+1

K,σ ),‖ut‖L1(Q), and on ‖∇u‖L2(Q).


5.5. Continuous entropy inequalities. From the discrete entropy inequalitieswe deduce continuous approximate entropy inequalities. The following theorem iscentral in the proof of the convergence of the scheme.

Theorem 5.1 (continuous approximate entropy inequalities). Assume hypothe-ses (H1), (H2), (H3), (H4), and (H5). Let D be an admissible discretization of Q, andlet uD be the corresponding approximate solution defined above. Then uD satisfies thefollowing approximate entropy inequalities: for all κ ∈ R, for all ψ ∈ C∞(R+ × R

d)such that ψ ≥ 0 and (ϕ(u)− ϕ(κ))

±ψ = 0 a.e. on Σ,∫

Q

η±κ (uD)ψt+Φ±κ (t, x, uD)·∇ψ+η±ϕ(κ)(ϕ(uD))∆ψ dxdt−

∫Σ

η±ϕ(κ)(ϕ(u))∇ψ ·ndγ(x)dt

+

∫Ω

η±κ (u0)ψ(0)dx+M

∫Σ

η±κ (u)ψ dγ(x)dt ≥ −E±D(ψ).(50)

Also assume that a uniform CFL condition δtn ≤ Csize(T ) for all n holds true (witha CFL number C that can be as large as desired). Then, for a given ψ, E±D(ψ) tendsto zero when the size of the discretization tends to zero.

Proof. The proof of Theorem 5.1 is quite similar to the proof of Theorem 5.1 in[EGHM02], except for the boundary terms, which require extra care. We will thereforestress the analysis of these terms and make reference to [EGHM02] when needed. Ofcourse, we can also limit ourselves to giving the proof of (50) when the nonnegativeKruzhkov entropy pairs are under consideration.

Let κ ∈ R, and let ψ ∈ C∞(R+ × Rd) be a nonnegative function satisfying

(ϕ(u) − ϕ(κ))+ψ = 0 a.e. on Σ. We define discrete values of ψ with respect tothe mesh as

Ψ0K = ψ(0, xK) ∀K ∈ T ,

Ψn+1K =

1

δtn

∫ tn+1

tnψ(t, xK)dt ∀K ∈ T ,∀n ∈ [[0, N ]],

ψn+1σ =

1

δtn

∫ tn+1

tnψ(t, xσ)dt ∀σ ∈ Eext,∀n ∈ [[0, N ]]

and set Ψn+1K,σ = Ψ

n+1L if σ = K|L and Ψn+1

K,σ = ψn+1σ if σ ∈ Eext,K .

The definition of the numerical flux Φ+,n+1K,σ,κ (see (48)) ensures that it is a conser-

vative flux, consistent with the function Φ+κ . Therefore, we have∑

σ∈EK

m(σ)Φ+,n+1K,σ,κ (U

n+1K , Un+1

K ) = 0 ∀K ∈ T , n ∈ [[0, N ]],

and the discrete entropy inequality (47) can then be rewritten as

η+κ (U

n+1K )− η+

κ (UnK)

δtnm(K)+

∑σ∈EK

m(σ)(Φ+,n+1K,σ,κ (U

n+1K , Un+1

Kσ)− Φ+,n+1

K,σ,κ (Un+1K , Un+1

K ))

−∑σ∈EK

τσ(η+ϕ(κ)(ϕ(U

n+1Kσ

))− η+ϕ(κ)(ϕ(U

n+1K ))

) ≤ 0.(51)

Multiplying (51) by δtnΨn+1K and summing over K ∈ T and n ∈ [[0, N ]] yields

A1 +A2 +A3 ≤ 0,


where

A1 =N∑n=0

∑K∈T

m(K)(η+κ (U

n+1K )− η+

κ (UnK))Ψ

n+1K ,

and, summing over the edges, A2 = A2int +A2ext, with

A2int =

N∑n=0

δtn∑K∈T

1

2

∑σ∈Eint,K

m(σ)(Ψn+1K (Φ+,n+1


Kσ)− Φ+,n+1


K ))

−Ψn+1K,σ (Φ

+,n+1K,σ,κ (U

n+1K , Un+1

Kσ)− Φ+,n+1

K,σ,κ (Un+1Kσ

, Un+1Kσ

)))

and

A2ext =

N∑n=0

δtn∑K∈T

∑σ∈Eext,K

m(σ)Ψn+1K

(Φ+,n+1K,σ,κ (U

n+1K , Un+1

Kσ)− Φ+,n+1


K )).

Similarly, A3 admits the decomposition A3 = A3int +A3ext, with

A3int =N∑n=0

δtn∑K∈T

1

2

∑σ∈Eint,K

τσ(η+ϕ(κ)(ϕ(U

n+1K ))− η+

ϕ(κ)(ϕ(Un+1Kσ

)))(Ψn+1

K −Ψn+1K,σ )

and

A3ext =N∑n=0

δtn∑K∈T

∑σ∈Eext,K

τσ(η+ϕ(κ)(ϕ(U

n+1K ))− η+

ϕ(κ)(ϕ(Un+1Kσ

)))Ψn+1K .

Now, set

I1 = −∫Q

η+κ (uD)ψt dx dt−

∫Ω

η+κ (u0)ψ(0, x) dx,

I2 = −∫Q

Φ+κ (t, x, uD) · ∇ψ dx dt−M

∫Σ

η+κ (u)ψ dγ(x)dt,

I3 = −∫Q

η+ϕ(κ)(ϕ(uD))∆ψ dx dt+

∫Σ

η+ϕ(κ)(ϕ(u))∇ψ · n dγ(x) dt .

We aim at proving the estimate I1+ I2+ I3 ≤ E+D(ψ) and, to that purpose, compare

I1 to A1, I2 to A2, and I3 to A3, respectively.A discrete integration by parts leads to |I1 − A1| ≤ E1,D(ψ), with E1,D(ψ) → 0

as size(D)→ 0 (see [EGHM02]).Using integration by parts in I2 and the fact that uD is piecewise constant, we

obtain

I2 = I2int + I2ext,

where I2ext is the boundary term and I2int gathers the sums on the internal edges.Precisely, we have

I2int = −N∑n=0

∑K∈T

1

2

∑σ∈Eint,K

(∫ tn+1

tn

∫σ

Φ+κ (t, x, U

n+1K ) · nK,σψ dγ(x) dt

−∫ tn+1

tn

∫σ

Φ+κ (t, x, U

n+1Kσ

) · nK,σψ dγ(x) dt

)


and

I2ext = −N∑n=0

∑K∈T

∑σ∈Eext,K

∫ tn+1

tn

∫σ

Φ+κ (t, x, U

n+1K ) · nK,σψdγ(x)dt−M

∫Σ

η+κ (u)ψdγ(x)dt.

As in [EGHM02], we prove |I2int−A2int| ≤ E int2,D(ψ), with E int

2,D(ψ)→ 0 as size(D)→0.

The comparison of I2ext with A2ext involves a term corresponding to the consis-tency error, and three terms related to the approximation of the boundary data:

I2ext−A2ext ≤ Ec1,ext2,D (ψ) + Eb1,ext

2,D (ψ) + Eb2,ext2,D (ψ)− T b2,ext

2,D (ψ),

where

Ec1,ext2,D (ψ) =

N∑n=0

∑K∈T

∑σ∈Eextκ

∣∣∣∣∣∫ tn+1

tn

∫σ

(Ψn+1K − ψ)Φ+

κ (·, ·, Un+1K ) · nK,σ dγ(x) dt

∣∣∣∣∣ ,

Eb1,ext2,D (ψ) =

N∑n=0

δtn∑K∈T

∑σ∈Eextκ

m(σ)|(Ψn+1K − ψn+1

σ )Φ+,n+1K,σ,κ (U

n+1K , Un+1

Kσ)|,

and

Eb2,ext2,D (ψ) =M

N∑n=0

∑K∈T

∑σ∈Eextκ

∣∣∣∣∣∫ tn+1

tn

∫σ

(u− κ)+ψ dγ(x) dt

− δtnm(σ)(Un+1Kσ

− κ)+ψn+1σ

∣∣∣∣∣are three terms converging to zero when size(D)→ 0 and

T b2,ext2,D (ψ) =

N∑n=0

δtn∑K∈T

∑σ∈Eextκ

m(σ)ψn+1σ

(Φ+,n+1K,σ,κ (U

n+1K , Un+1

Kσ) +M(Un+1

Kσ− κ)+

).

From the definition of Φ+,n+1K,σ,κ (see (48)) and from the monotony of the scheme,

Φ+,n+1K,σ,κ (a, b) = Fn+1

K,σ (aκ, bκ)− Fn+1K,σ (κ, κ) ≥ −Lip(Fn+1

K,σ )(b− κ)+

follows, and this entails T b2,ext2,D (ψ) ≥ 0 .

Now, to compare I3 to A3 we make the distinction between the different contribu-tions of the terms (inside and on the boundary of Ω). Indeed, since the approximatesolution uD is piecewise constant, the term I3 reads as I3 = I3int + I3ext, where

I3int =

N∑n=0

∑K∈T

1

2

∑σ∈Eintκ

(η+ϕ(κ)(ϕ(U

n+1Kσ

))− η+ϕ(κ)(ϕ(U

n+1K )))

∫ tn+1

tn

∫σ

∇ψ · nK,σ dγ(x) dt

and

I3ext =

N∑n=0

∑K∈T

∑σ∈Eextκ

∫ tn+1

tn

∫σ

(η+ϕ(κ)(ϕ(u))− η+

ϕ(κ)(ϕ(Un+1K )))∇ψ · nK,σ dγ(x) dt.


A consistency error term controls the proximity of A3int to I3int:

|I3int−A3int| ≤ Ec,int3,D (ψ),

with Ec,int3,D (ψ)→ 0 when size(D)→ 0 [EGHM02].

In order to compare I3ext and A3ext, rearrange the term I3ext, up to consistencyor approximation errors, to get

I3ext ≤N∑n=0

δtn∑K∈T

∑σ∈Eextκ

τσ

(η+ϕ(κ)(ϕ(U

n+1Kσ

))− η+ϕ(κ)(ϕ(U

n+1K ))

)(Ψn+1

K,σ −Ψn+1K )

+ Ec,ext3,D (ψ) + Eb1,ext

3,D (ψ),

where

Ec,ext3,D (ψ)

=N∑n=0

∑K∈T

∑σ∈Eextκ

2 maxu∈[A,B]

η+ϕ(κ)(ϕ(u))

∣∣∣∣∣∫ tn+1

tn

∫σ

(∇ψ · n− ψn+1

σ −Ψn+1K

dK , σ

)dγ(x) dt

∣∣∣∣∣ ,Eb1,ext3,D (ψ) =

N∑n=0

∑K∈T

∑σ∈Eextκ

∫ tn+1

tn

∫σ

∣∣ϕ(u)− ϕ(Un+1σ )

∣∣∣∣∇ψ · n∣∣ dγ(x) dt.Then we have

I3ext−A3ext =

N∑n=0

δtn∑K∈T

∑σ∈Eextκ

τσ(η+ϕ(κ)(ϕ(U

n+1Kσ

)) − η+ϕ(κ)(ϕ(U

n+1K )))Ψn+1

K,σ

+ Ec,ext3,D (ψ) + Eb1,ext

3,D (ψ) .

Now, either η+ϕ(κ)(ϕ(U

n+1Kσ

)) = 0, and in that case

(η+ϕ(κ)(ϕ(U

n+1Kσ

))− η+ϕ(κ)(ϕ(U

n+1K ))Ψn+1

K,σ ) ≤ 0,

or η+ϕ(κ)(ϕ(U

n+1Kσ

)) > 0. In the latter case, the condition (ϕ(u)− ϕ(κ))+ψ = 0 a.e. on

Σ ensures that there exists (t, x) ∈ [tn, tn+1]× σ such that ψ(t, x) = 0. Consequently,we have

Ψn+1K,σ ≤ Lip(ψ)(δtn + diam(σ)) .

This estimate, combined with the inequality

η+ϕ(κ)(ϕ(U

n+1Kσ

))− η+ϕ(κ)(ϕ(U

n+1K )) ≤ (η+

ϕ(κ))′(ϕ(Un+1

σ ))(ϕ(Un+1σ )− ϕ(Un+1

K )),

which is consequence of the convexity of the function η+ϕ(κ), leads to

I3ext−A3ext ≤ Eb2,ext3,D (ψ) + Ec,ext

3,D (ψ) + Eb1,ext3,D (ψ),

where

Eb2,ext3,D (ψ) =

N∑n=0

δtn∑K∈T

∑σ∈Eextκ

τσLip(ψ)(δtn + diam(σ))|ϕ(Un+1σ )− ϕ(Un+1

K )| .


Using the Cauchy–Schwarz inequality, together with the L2(0, T ;H10 (Ω)) estimate of

Proposition 5.2 and the inequality ϕ(a)− ϕ(b) ≤√Lip(ϕ)(ζ(a)− ζ(b)), yields

Eb2,ext3,D (ψ) ≤ C

N∑n=0

δtn∑K∈T

∑σ∈Eextκ

τσ(δtn + diam(σ))2 .

Therefore, a simple way to ascertain that Eb2,ext3,D (ψ) converges to zero is to suppose a

uniform CFL condition such as δtn ≤ Csize(T ) for all n (where the CFL number Ccan be as large as desired). Then we conclude the proof of Theorem 5.1 by defining

E+D(ψ) as the sum of the errors E1,D(ψ), E int

2,D(ψ), Ec1,ext2,D (ψ), Eb1,ext

2,D (ψ), Eb2,ext2,D (ψ),

Ec,int3,D (ψ), Ec,ext

3,D (ψ), Eb1,ext3,D (ψ), and Eb2,ext

3,D (ψ).

5.6. Convergence of the scheme. Let Dn be a sequence of discretizations,such that size(Dn) tends to zero. We wish to prove the convergence of uDn to anentropy solution of problem (1). For that purpose, in view of the uniqueness Theorem4.1, it suffices to show that, up to a subsequence, uDn

tends in the nonlinear weak-sense to an entropy process solution of (1). We obtain compactness properties usingestimates on uDn derived from discrete estimates on UDn , then pass to the limit ininequality (50).

5.6.1. Nonlinear weak- compactness. The maximum principle ensures that(uDn

) is bounded in L∞(Q). Consequently, there exist u ∈ L∞(Q× (0, 1)) such that,up to a subsequence, uDn tends to u in the nonlinear weak- sense.

5.6.2. Compactness in L2(Q). From discrete estimates obtained in Propo-sition 5.2 we easily deduce (see, e.g., [EGH00]) the following inequalities on zD =ζ(uD)− ζ(uD).

Proposition 5.3 (space translation estimates). Assume hypotheses (H1), (H2),(H3), (H4), and (H5). There exists a constant C1 such that

∀y ∈ Rd,

∫ T

0

∫Ωy

(zD(t, x+ y)− zD(t, x))2dxdt ≤ C1|y|(|y|+ size(T )),

where Ωy = x ∈ Ω, [x, x+ y] ⊂ Ω.The hypothesis (H5) includes the assumption ut ∈ L1(Q), while the discrete

evolution equation (43) relates the discrete time derivative of uD to its discrete spacederivative. Therefore the following time translation estimate on zD is available.

Proposition 5.4 (time translation estimates). Assume hypotheses (H1), (H2),(H3), (H4), and (H5). There exists a constant C2 such that

∀s > 0,∫ T−s

0

∫Ω

(zD(t+ s, x)− zD(t, x))2dxdt ≤ C2 s.

Since the function zD vanishes on Σ, it can be extended by zero out of Q. Thenusing the Frechet–Kolmogorov theorem (see, e.g., [Bre83]), we get the existence ofa function z ∈ L2(0, T,H1(Ω)) such that, up to a subsequence, zDn → z in L2(Q).Besides, since zD = ζ(uD)− ζ(uD) and ζ(uD) converges to ζ(u) in L

2(Q), we get theconvergence of ζ(uDn) in L

2(Q) (to ζ(u)+z). On the other hand, the nonlinear weak- convergence of (uDn

) shows that ζ(uDn) converges also to ζ(u) weakly in L∞(Q), so

that ζ(u)+z = ζ(u). In particular, ζ(u) does not depend on the last argument α, andthe trace of ζ(u) on Σ is ζ(u). See [EGHM02] for more details on this step of the proof.


5.6.3. Conclusion. It remains to pass to the limit in the continuous entropyinequalities to prove that u is an entropy process solution. The uniqueness Theorem4.1 proves that u does not depend on α and is the unique entropy weak solution ofproblem (1). Besides, the whole sequence uDn is convergent (u is the unique possiblelimit), and by definition of the nonlinear weak- convergence, (uDn

)2 also convergesweakly to (u)2 so that uDn converges to u in L2(Q) (strong), and in all Lp(Q), for1 ≤ p < +∞. Therefore, we have proved the following theorem.

Theorem 5.2. Let Dn be a sequence of discretizations, such that size(Dn) tendsto zero. Assume hypotheses (H1), (H2), (H3), (H4), (H5), and (H6) (or (H6bis)).Then, for every 1 ≤ p < +∞, (uDn) converges to the unique entropy solution ofproblem (1) in Lp(Q).

Acknowledgments. We warmly thank R. Eymard, M. Ohlberger, A. Porretta,and T. Gallouet for wise advice and fruitful discussions.

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